Abstract
In this paper, we introduce and investigate new subclasses (Yamakawa-type bi-starlike functions and another class of Lashin, both mentioned in the reference list) of bi-univalent functions defined in the open unit disk, which are associated with the Gegenbauer polynomials and satisfy subordination conditions. Furthermore, we find estimates for the Taylor–Maclaurin coefficients |a2| and |a3| for functions in these new subclasses. Several known or new consequences of the results are also pointed out.
Highlights
Introduction and PreliminariesIn geometric function theory, there have been numerous interesting and fruitful usages of a wide variety of special functions, q-calculus and special polynomials; for example, the Fibonacci polynomials, the Faber polynomials, the Lucas polynomials, thePell polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the second kind
In the above Remarks 3 and 4, by fixing λ = 1 and λ =, we obtain the new estimates of | a2 | and | a3 | for the function classes Y S ∗Σ Φλand NΣ Φλrelated with Chebyshev polynomials and Legendre polynomials, respectively
Yamakawa-type bi-starlike functions related with the Gegenbauer polynomials are defined for the first time, and initial Taylor coefficients and Fekete–Szegő inequality are obtained
Summary
Introduction and PreliminariesIn geometric function theory, there have been numerous interesting and fruitful usages of a wide variety of special functions, q-calculus and special polynomials; for example, the Fibonacci polynomials, the Faber polynomials, the Lucas polynomials, thePell polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the second kind. The generating function of Gegenbauer polynomials is given by (see [1,4]) The importance of the class TR (λ), λ > 0, follows as well from the paper of Hallenbeck [9], who studied the extreme points of some families of univalent functions and proved that
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